Application of Derivatives

Find points at which the tangent to the curve y = x 3 – 3x 2 – 9x + 7 is parallel to the x-axis. - Wired Faculty

Question

Find points at which the tangent to the curve y = x³ – 3x² – 9x + 7 is parallel to the x-axis.

Solution

The equation of curve is
$straight y equals straight x cubed minus 3 straight x squared minus 9 straight x plus 7$
$therefore space space space space space space space space dy over dx space equals space 3 straight x squared minus 6 straight x minus 9$
For the points on the curve, where tangents are parallel to x-axis
$dy over dx space equals space 0 space space space space space rightwards double arrow space space space space 3 straight x squared minus 6 straight x minus 9 space equals space 0 space space space space space rightwards double arrow space space space straight x squared minus 2 straight x minus 3 space equals space 0$
$therefore space space space space left parenthesis straight x minus 3 right parenthesis thin space left parenthesis straight x plus 1 right parenthesis space equals space 0 space space space space space space rightwards double arrow space space space space straight x space equals space 3 comma space space minus 1$
When $straight x space equals 3 comma space space space straight y equals left parenthesis 3 right parenthesis cubed minus 3 left parenthesis 3 right parenthesis squared space minus space 9 left parenthesis 3 right parenthesis plus space 7 space equals space 27 minus 27 minus 27 plus 7 space equals space minus 20$
When $straight x equals negative 1 comma space space straight y space equals space left parenthesis negative 1 right parenthesis cubed minus 3 left parenthesis negative 1 right parenthesis squared minus 9 left parenthesis negative 1 right parenthesis plus 7 space equals negative 1 minus 3 plus 9 plus 7 space equals space 12$
$therefore space space space space space space required space points space are space left parenthesis 3 comma space minus 20 right parenthesis comma space left parenthesis negative 1 comma space 12 right parenthesis.$

For Daily Free Study Material Join wiredfaculty